A remark on positively curved 4-manifolds (Q1325953)

This paper is inspired by Hopf's conjecture on \(S^ 2\times S^ 2\) which states that \(S^ 2\times S^ 2\) does not admit a Riemannian metric of positive sectional curvature. Up to now this conjecture still remains unsolved. Using the Bochner technique for the Laplace operator on 2-forms the following theorem is shown: Theorem. Let \(M\) be an oriented connected compact 4-manifold with indefinite intersection form (e.g. \(M= S^ 2\times S^ 2\)). Then there is no Riemannian metric on \(M\) such that (i) the sectional curvature satisfies \(K\geq 1\), (ii) the covariant differential of the curvature tensor satisfies \(|\nabla R|\leq{2\over \pi}\).